Question: Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{x^2 + 2x}{x^2 - 2x - 8}$
First factor the expressions in the numerator and denominator. $ \dfrac{x^2 + 2x}{x^2 - 2x - 8} = \dfrac{(x)(x + 2)}{(x - 4)(x + 2)} $ Notice that the term $(x + 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x + 2)$ gives: $t = \dfrac{x}{x - 4}$ Since we divided by $(x + 2)$, $x \neq -2$. $t = \dfrac{x}{x - 4}; \space x \neq -2$